Assigning hotkey to set keyframes without using the menu
Finally figured it out!

Go to "Preferences -> Keymap" and search for "insert keyframe".

Chose which view you want to edit the hotkey for. For example, under "Pose" click the triangle and it'll show you the python command "anim.keyframe_insert_menu"



Simply change it to "anim.keyframe_insert" and press Enter.

There's one more thing to do. If you go into Pose Mode now, select a bone and press "i" - you won't get the menu anymore BUT you'll get an error "No active keying set".

Under the python code is a confusingly greyed out drop-down menu. Click that and choose what you want the "i" shortcut to do, in my case I chose "Whole Character"



That's it! Now when in Pose Mode selecting one controller and pressing "i" now instantly keys everything on the rig!